The Unarmed Fighting style from Tasha's states:
Your unarmed strikes can deal bludgeoning damage equal to 1d6 + your Strength modifier. If you strike
with two free hands, the d6 becomes a d8. When you successfully start a grapple, you can deal 1d4
bludgeoning damage to the grappled creature. Until the grapple ends, you can also deal this damage to
the creature whenever you hit it with a melee attack.
Would a, say, 2nd-level Monk gain a benefit from taking a single level in Fighter, making unarmed strikes with a d8+STR?
This seems better than the 1d4 from the early levels.
Best Answer
This works, but it becomes mostly redundant at higher levels.
You have the right of it - by taking this fighting style, you can use a d6 (or d8) for your unarmed strikes at the lower levels of monk before your martial arts die matures. Your martial arts die becomes d6 at 5th level monk and d8 at 11th level monk, so the fighting style loses some of its utility at each of these levels.
In my estimation, this is not worth a 1 level multiclass dip.